Probability spaces where the state space is finite or countable, are easy to describe and have many applications. This chapter begins with a presentation of the most common examples. The main part of the chapter is devoted to the rigorous definition of the Lebesgue measure and the Gaussian measure on the real line. This requires the introduction of the Borel \(\sigma \) -algebra on the real line. To obtain the existence of the Lebesgue measure and the Gaussian measure the extension theorem is used, which provides a unique extension of a suitable pre-measure on an algebra to a measure on the \(\sigma \) -algebra generated by the algebra.